(i) That \(\omega (Jv,w)\)  is symmetric is equivalent to
\(J \in \mathfrak {sp}(\mathcal {H})\) , and as \(J^{-1} = -J\)  characterizes complex structures,
(i) follows.
(ii) We have seen in (REF )
that \(I\)  is \(\omega \) -positive.
Let \(J\)  be another \(\omega \) -positive complex structure.
Writing \(J = u e^x\)  according to the polar
decomposition of \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  with
\(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\)  and \(x\in {\mathfrak {p}}\)  ([1]}),
we see that \(J^2 = -\mathbf {1}\)  is equivalent to \(u^2 = -\mathbf {1}\) 
(\(u\)  is a complex structure) and \(ux=-xu\)  (\(x\)  is antilinear
with respect to \(u\) ). If this is the case, then
\(\omega (Jv,v) = \omega (u e^x v, v)= \omega (e^{-x/2}u e^{x/2}v,v)= \omega (u e^{x/2}v,e^{x/2}v) \) 
shows that \(J\)  is \(\omega \) -positive if and only if
\(u\)  has this property. Since \(u\)  is complex linear,
\(\mathcal {H}\)  decomposes into \(\pm i\) -eigenspaces
\(\mathcal {H}_\pm \)  of \(u\) . Then \(\omega (uv,v)\)  is positive definite on
\(\mathcal {H}_+\)  and negative definite on \(\mathcal {H}_-\) , so that the
\(\omega \) -positivity implies \(u = I\) . This proves (ii).
(iii) For \(g = u e^x \in \mathop {{\rm Sp}}\nolimits (\mathcal {H})\) , the relation
\(g^{-1}Ig = e^{-x} I e^x = I e^{2x}\) 
shows that \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  acts transitively on
\({\mathcal {I}}_\omega \) . As \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  is the stabilizer of \(I\) ,
(iii) follows from the smoothness of the map
\(J = I e^x \mapsto x = {\textstyle {\frac{1}{2}}}\log (J^\top J)\) .
(iv) Let \((v,w)_A := \omega (Av,w)\)  denote the real Hilbert
structure on \(\mathcal {H}\)  defined by \(A\) . Then
\( \omega (x,y) = \omega (A(A^{-1}x), y) = (A^{-1}x,y)_A \) 
implies that \(A^{-1}\)  is skew-symmetric with respect to
\((\cdot ,\cdot )_A\) , and the same holds for \(A\)  itself.
Therefore \(-A^2 \ge 0\)  and
\(J := (-A^2)^{-1/2}A\)  is a complex structure leaving
\((\cdot ,\cdot )_A\)  invariant (cf. [2]}).
Then
\( \omega (Jv,v)= \omega ((-A^2)^{-1/2}Av,v)= ((-A^2)^{-1/2}v,v)_A \) 
is positive definite, so that \(J\)  is \(\omega \) -positive.
If \(J^{\prime }\)  is another \(\omega \) -positive complex structure commuting
with \(A\) , then the construction of \(J\)  shows that it also commutes
with \(J\) . Therefore \(JJ^{\prime }\)  is an involution and since
\(J\)  and \(J^{\prime }\)  are \(\omega \) -positive, the \(-1\) -eigenspace of this
involution is trivial, which leads to \(J^{\prime } = J\) .
(v) From the polar decomposition
\(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) = \mathop {\rm U{}}\nolimits (\mathcal {H}) \exp ({\mathfrak {p}}_2)\quad \mbox{ with } \quad {\mathfrak {p}}_2 := {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H})= \lbrace  x \in {\mathfrak {p}}\colon  \Vert x\Vert _2 < \infty \rbrace  \) 
we derive that
\({\mathcal {I}}_\omega ^{\rm res} = I e^{{\mathfrak {p}}_2}.\) 
For the entire function \(F(z) := \frac{e^z - 1}{z}\) , we then have
\( Ie^x - I = I F(x) x, \) 
Since \(F(x)\)  is invertible for the real symmetric operator
\(x\)  by the Spectral Mapping Theorem, it follows that
\(Ie^x - I\)  is Hilbert–Schmidt if and only if \(x\) 
has this property.
For \(J = I e^x\)  we also observe that
\([I,J] = -e^x - Ie^x I = e^{-x} - e^{x}= e^{-x}(\mathbf {1} - e^{2x})= - e^{-x} F(2x) 2 x, \) 
so that \([I,J]\)  is Hilbert–Schmidt if and only if \(x\)  is.
This leads to \({\mathcal {I}}_\omega ^{\rm res} = {\mathcal {I}}_\omega \cap \mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
The following theorem is a useful tool when dealing with invariant
cones in symplectic Lie algebras.
Theorem 6.7   For the canonical open invariant cones
in \(\mathfrak {sp}(\mathcal {H})\) , \(\mathfrak {sp}_{\rm res}(\mathcal {H})\)  and \(\mathfrak {hsp}(\mathcal {H})\)  we have the following
conjugacy results:
(i)
\(W_{\mathfrak {sp}(\mathcal {H})} = \mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits (\mathcal {H})) C_{{\mathfrak {u}}(\mathcal {H})}\)  and
\(C_{{\mathfrak {u}}(\mathcal {H})} = W_{\mathfrak {sp}(\mathcal {H})} \cap {\mathfrak {u}}(\mathcal {H})\) .
(ii)
\(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})} := W_{\mathfrak {sp}\mathcal {H})} \cap \mathfrak {sp}_{\rm res}(\mathcal {H}) = \mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})) C_{{\mathfrak {u}}(\mathcal {H})}\) .

(iii)
\(W_{\mathfrak {hsp}(\mathcal {H})} = \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits (\mathcal {H}))({\mathbb {R}}\times \lbrace 0\rbrace  \times C_{{\mathfrak {u}}(\mathcal {H})})\) 
for the corresponding Lie group \(\mathop {\rm HSp{}}\nolimits (\mathcal {H}) = \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \mathop {{\rm Sp}}\nolimits (\mathcal {H})\) .
(iv)
\(W_{\mathfrak {hsp}_{\rm res}(\mathcal {H})} :=W_{\mathfrak {hsp}(\mathcal {H})} \cap \mathfrak {hsp}_{\rm res}(\mathcal {H})= \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H}))({\mathbb {R}}\times \lbrace 0\rbrace  \times C_{{\mathfrak {u}}(\mathcal {H})})\) 
for \(\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H}) := \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .


